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A priori Convergence Theory for Reduced-Basis
Approximations of Single-Parametric Elliptic Partial
Differential Equations
Yvon Maday, Anthony Patera, Gabriel Turinici
To cite this version:
Yvon Maday, Anthony Patera, Gabriel Turinici. A priori Convergence Theory for Reduced-Basis
Approximations of Single-Parametric Elliptic Partial Differential Equations. Journal of Scientific
Computing, Springer Verlag, 2002, 17 (1-4), pp.437-446. �hal-00798410�
A Priori Convergence Theory for
Reduced-Basis Approximations of
Single-Parameter Elliptic Partial Differential
Equations
Yvon MADAY
∗, Anthony T. PATERA
†, Gabriel TURINICI
‡Abstract
We consider “Lagrangian” reduced-basis methods for single-parameter symmetric coercive elliptic partial differential equations. We show that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced–basis approximation converges exponentially to the ex-act solution uniformly in parameter space. Furthermore, the conver-gence rate depends only weakly on the continuity-coercivity ratio of the operator: thus very low-dimensional approximations yield accu-rate solutions even for very wide parametric ranges. Numerical tests (reported elsewhere) corroborate the theoretical predictions.
1
Introduction
The development of computational methods that permit rapid and reliable evaluation of the solution of partial differential equations in the limit of many queries is relevant within many design, optimization, control, and characterization contexts. One particular approach is the reduced-basis method, first introduced in the late 1970s for nonlinear structural anal-ysis [1, 9], and subsequently developed more broadly in the 1980s and 1990s [3, 5, 10, 13, 2]. The reduced-basis method recognizes that the field variable is not, in fact, some arbitrary member of the infinite-dimensional space associated with the partial differential equation; rather, it resides, or
∗Laboratoire d’Analyse Num´erique, Universit´e Pierre et Marie Curie, Boˆıte courrier
187, 75252 Paris Cedex 05, France; Email: maday@ann.jussieu.fr
†Department of Mechanical Engineering, M.I.T., 77 Mass. Ave., Cambridge, MA,
02139, USA; Email: patera@mit.edu
‡ASCI-CNRS Orsay, and INRIA Rocquencourt M3N, B.P. 105, 78153 Le Chesnay
“evolves,” on a much lower-dimensional manifold induced by the paramet-ric dependence.
Let Y be an Hilbert space with inner product and norm (· , ·)Y and
k · kY = (· , ·)1/2Y , respectively. Consider a parametrized “bilinear” form
a : Y × Y × D → R, where D ≡ [0, µmax], and a bounded linear form
f : Y → R. We introduce the problem to be solved: Given µ ∈ D, find u∈ Y such that
a(u(µ), v; µ) = f (v), ∀ v ∈ Y . (1) Under natural conditions on the bilinear form a (e.g. continuity and coer-civity) it is readily shown that this problem admits a unique solution.
We introduce an approximation index N , the parameter sample SN =
{α1, . . . , αN}, and the solutions u(αk), k = 1, . . . , N , of problem (1) for
this set of parameters. We next define the reduced-basis approximation space
WN = span{u(αk), k = 1, . . . , N}.
Our reduced-basis approximation is then: Given µ∈ D, find uN(µ)∈ WN
such that
a(uN(µ), v; µ) = f (v), ∀ v ∈ WN . (2)
This discrete problem is well posed as well under the same former continuity and coercivity conditions.
The reduced-basis approach, as earlier developped, is typically local in parameter space in both practice and theory. To wit, the αk are chosen in
the vicinity of a particular parameter point µ∗ and the associated a
pri-oriconvergence theory relies on asymptotic arguments in sufficiently small neighborhoods of µ∗ [5]. In this paper we present, for single-parameter
symmetric coercive elliptic partial differential equations, a first theoretical a priori convergence result that demonstrates exponential convergence of reduced-basis approximations uniformly over an extended parameter do-main. The proof requires, and thus suggests, a point distribution in param-eter space which does, indeed, exhibit superior convergence properties in a variety of numerical tests [15]. We refer to [16, 17] for a different analysis within the homogeneization framework.
2
Problem Formulation
Let us define the parametrized “bilinear” form a : Y × Y × D → R as a(w, v; µ)≡ a0(w, v) + µa1(w, v) , (3)
where the bilinear forms a0: Y×Y → R and a1: Y×Y → R are continuous,
symmetric and positive semi-definite; suppose moreover that a0is coercive,
inducing a (Y -equivalent) norm ||| · |||2 = a
0(· , ·). It follows from our
assumptions that there exists a real positive constant γ1 such that
0≤a1(v, v) a0(v, v) ≤ γ1
, ∀ v ∈ Y . (4)
For the hypotheses stated above, it is readily demonstrated that the prob-lem (1) has a unique solution.
Many situations may be modeled by our rather simple problem state-ment (1), (3). For example, if we take Y = H1
0(Ω) where Ω is a smooth
bounded subdomain of Rd=2, and set a
0(w, v) =RΩ∇w · ∇v, a1=RΩwv,
we model conduction in thin plates; here µ represents the convective heat transfer coefficient. If we take Y = H1
0(Ω) for Ω ⊂ Rd=1,2, or 3, with
Ω1⊂ Ω (Ω1, Ω bounded and sufficiently regular), and set a0=RΩ∇w · ∇v,
a1=RΩ1∇w · ∇v, we model variable-property heat transfer; here 1 + µ is
the ratio of thermal conductivities in domains Ω\Ω1and Ω1. Other choices
of a0 and a1 can model variable rectilinear geometry, variable orthotropic
properties, and variable Robin boundary conditions.
The space Y is typically of infinite dimension so u(µ) is, in general, not exactly calculable. In order to construct our reduced-basis space WN,
we must therefore replace u(µ)∈ Y by a “truth approximation” uN
(µ)∈ YN ⊂ Y , where uN
is the Galerkin approximation satisfying a(uN(µ), v; µ) = f (v), ∀ v ∈ YN . Here YN, of finite (but typically very high) dimension
N , is a sufficiently rich approximation subspace such that|||u(µ)−uN(µ)
||| is sufficiently small for all µ inD; for example, for Y = H1
0(Ω) we know that, for any desired ε >
0, we can indeed construct a finite-element approximation space, YN (ε),
such that|||u(µ) − uN (ε)(µ)||| ≤ ε.
It shall prove convenient in what follows to introduce a generalized eigenvalue problem: Find (ϕN
i ∈ YN, λNi ∈ R), i = 1, . . . , N , satisfying
a1(ϕNi , v) = λ N
i a0(ϕNi , v), ∀ v ∈ Y N
. We shall order the (perforce real, non-negative) eigenvalues as 0≤ λN N ≤ λ N N −1≤ · · · ≤ λ N 1 ≤ γ1, where the
last inequality follows directly from (4). We may choose our eigenfunctions such that
a0(ϕNi , ϕ N
j ) = δi j , (5)
and hence a1(ϕNi , ϕNj ) = λNi δi j, where δi j is the Kronecker-delta symbol;
and such that YN can be expressed as span
{ϕi, i = 1, . . . ,N }. Note that,
(the complications associated with) a continuous spectrum — and, as we shall see, at no loss in rigor.
We conclude this section by noting that uN(µ) can be expressed as
uN(µ) = N X i=1 fiNϕ N i 1 + µλN i , (6) where fN i = f (ϕ N i ).
3
A Priori Convergence Theory
We propose here to choose the sample points αk, k = 1, . . . , N ,
log-equidistributed inD, in the sense that, if we set δN = ln(γµmax+ 1)/N ,
and γ is any finite upper bound for γ1 1, then
αk = exp{− ln γ + k X ℓ=1 ˜ δℓN} − γ−1,
where we assume that there exists a constant c∗such that
˜ δkN
δN ≤ c ∗
, ∀k, k = 1, . . . , N
and also thatPNℓ=1δ˜ℓN = ln(γµmax+ 1).
Denote the reduced-basis approximation space as WN
N = span{u N(α
k), k =
1, . . . , N}. Although in general dim(WN
N ) ≤ N, we can suppose that
dim(WN
N ) = N (otherwise we eliminate elements from WNN until it
con-tains only linearly independent vectors). Then, the (reduced basis) problem is : Given µ∈ D, find uN N(µ)∈ W N N such that a(uNN(µ), v; µ) = f (v), ∀ v ∈ W N N . (7)
This problem admits a unique solution. Our goal is to (sharply) bound |||uN(µ)
− uN
N(µ)|||, for all µ ∈ D, as
a function of N (and ultimately N as well). This error bound in the energy norm can be readily translated into error bounds on continuous-linear-functional outputs [12]; we do not consider this extension further here.
We shall need two standard results from the theory of Galerkin approx-imation of symmetric coercive problems [14]:
a(uN− uNN, u N − uNN; µ) = inf wN N∈W N N a(uN− wNN, u N − wNN; µ) ; (8) 1Note that γ
and
a(uN, uN; µ)≤ a(u, u; µ) . (9) From the definition of the||| · ||| norm, the positive semidefiniteness of a1,
(3), (4) and (8) we can write |||uN (µ)− uN N(µ)|||2≤ a(u N (µ)− uN N(µ), u N (µ)− uN N(µ), µ) ≤ inf wN N∈W N N a(uN(µ)− wNN, u N (µ)− wNN, µ) ≤ (1 + µmaxγ1) inf wN N∈W N N |||uN (µ)− wN N|||2, ∀ µ ∈ D. (10)
Also from the definition of the ||| · ||| norm and the positive semidefi-niteness of a1, (3), (4) and (9), we obtain
|||uN(µ)||| ≤ (1 + µmaxγ1)1/2|||u(µ)|||, ∀ µ ∈ D (11)
We begin with a preparatory result in Lemma 3.1 Let
g(z, λ) = 1 1−λ
γ + λez
(12)
for z∈ Z ≡ [ln(γ−1),∞] and λ ∈ Λ ≡ [0, γ] (recall γ is our strictly positive
upper bound for γ1). Then, for any q≥ 0
|Dq1g(z, λ)| ≤ 2qq! , ∀ z ∈ Z, ∀ λ ∈ Λ ,
where Dq1g denotes the qth-derivative of g with respect to the first argument.
Proof. We first remark that for any p≥ 0
0≤ gp(z, λ)≤ 1, ∀ z ∈ Z, ∀ λ ∈ Λ, , (13) where gp is the pth-power of g. This follows since
∀ z ∈ Z, ez
≥ γ−1, and
hence,∀ z ∈ Z, ∀ λ ∈ Λ, 1 − λ/γ + λez
≥ 1. We next claim that, for m≥ 2,
Dm−11 g(z, λ) = m
X
n=1
amn gn(z, λ) , (14)
where, for m≥ 1 (and a1 1≡ 1) am+11 = −am 1 am+1 n = −n amn + (n− 1) 1−λγam n−1, 2≤ n ≤ m am+1m+1 = m 1−λ γ am m. (15)
We prove this result by induction. We first differentiate g to obtain D11g(z, λ) = −λe z 1−λ γ + λez 2 = −1−λ γ + λe z+ 1 −λ γ 1−λ γ + λez 2 =−g(z, λ) + 1−λ γ g2(z, λ) ; (16)
(14) and (15) for m = 2 directly follows. We now differentiate (14) for m = m, and exploit (16), to obtain
D1(m+1)−1(z, λ) = m X n=1 am n n gn−1(z, λ) D11g(z, λ) = m X n=1 amn n gn−1(z, λ) −g(z, λ) + 1−λγ g2(z, λ) = m X n=1 −n am n gn(z, λ) + m+1X n=2 (n− 1) 1−λγ amn−1gn(z, λ) ;
(14) and (15) for m = m + 1 directly follows. Thus, (14), (15) are valid for m≥ 2, as required.
It now follows from (13) and (14) that, for any q≥ 1, |Dq1g(z, λ)| ≤ Sq,
where Sq = q+1P n=1|a
q+1
n |. We now invoke (15), and observe that ∀ λ ∈ Λ,
|1 − λγ| ≤ 1, to obtain Sq ≤ 2q Sq−1; since S1 ≤ 2, we conclude that
Sq ≤ 2qq!, and the lemma is thus proven.
We now prove a bound for the best approximation result in Lemma 3.2 For N ≥ Ncrit
inf wN N∈W N N |||uN(µ)− wNN||| ≤ |||u N (0)||| exp −N Ncrit , ∀ µ ∈ D ,
where Ncrit≡ c∗e ln(γ µmax+ 1).
Proof. To facilitate the proof, we shall effect a change of coordinates in parameter space. To wit, we let eD ≡ [ln γ−1, ln(µ
max+γ−1)], and introduce
τ : eD → D, as τ(˜µ) = eµ˜
− γ−1 so that τ−1(µ) = ln(µ + γ−1). We then set
˜
u(˜µ) = u(τ (˜µ)), ˜uN(˜µ) = uN(τ (˜µ)), and ˜uNN(˜µ) = u N N(τ (˜µ)). We note that ˜ uN(˜µ) = N X i=1 fiNϕ N i 1−λNi γ + λ N i eµ˜ = N X i=1 fiNϕ N i g(˜µ, λ N i ), (17)
from (6), our change of variable, and the definition (12).
We now observe that in our mapped coordinate, the sample points ˜αk≡
τ−1(αk), k = 1, . . . , N , are equi-distributed with separation ˜αk+1− ˜αk ≃
ln(γµmax+ 1)/N . It thus follows that, given any ˜µ∈ ˜D, we can construct
a closed interval eI∆µ˜˜ of length ˜∆, that includes ˜µ and Mµ˜( ˜∆, δN) distinct
points ˜αPµ˜ n, n = 1, . . . , M . Here M ˜ µ( ˜∆, δ N) is of the order of ˜ ∆ δN; more precisely, Mµ˜( ˜∆, δN)≥ ˜ ∆ c∗δ N . (18)
In what follows, we shall often abbreviate Mµ˜( ˜∆, δ
N) as M .
Now, for any ˜µ∈ eD, we introduce ˆuµ˜∈ WN
N given by ˆ uµ˜ ≡ M X n=1 e Qµ˜ n(˜µ) u N (τ (˜αPµ˜ n)) = M X n=1 e Qµ˜ n(˜µ) ˜u N (˜αP˜µ n) = M X n=1 e Qµn˜(˜µ) N X i=1 fiNϕ N i g(˜αPµ˜ n, λ N i ) ,
where the characteristic functions eQµ˜
n are uniquely determined by eQµn˜ ∈
PM −1(eI∆˜µ˜), n = 1, . . . , M , and eQµn˜(˜αPµ˜ n′
) = δnn′, 1 ≤ n, n′ ≤ M; here
PM −1(eI∆˜µ˜) refers to the space of polynomials of degree ≤ M − 1 over eI ˜ µ ˜ ∆. We thus obtain ˆ uµ˜= N X i=1 fiNϕ N i [eI ˜ µ M −1g(·, λ N i )] (˜µ) , (19)
where, for given λ, eIM −1µ˜ g(·, λ) is the (M − 1)th-order polynomial
inter-polant of g(·, λ) through the ˜αPµ˜
n, n = 1, . . . , M ; more precisely, eI ˜ µ M −1g(·, λ) ∈ PM −1(eI∆˜µ˜), and (eI ˜ µ M −1g(·, λ))(˜αPnµ˜) = g(˜αP ˜ µ n, λ), n = 1, . . . , M . Note that [eIM −1µ˜ g(·, λ)](τ−1(µ)) is not a polynomial in µ.
It now follows from (5), (6), (17) and (19) that |||˜uN (˜µ)− ˆuµ˜ ||| ≤ PNi=1 f N i ϕ N i g(˜µ, λNi )− [eI ˜ µ M −1g(·, λ N i )] (˜µ) ≤ supλ∈Λ |g(˜µ, λ) − [eI ˜ µ M −1g(·, λ)] (˜µ)| |||u N(0) ||| . (20) We next invoke the standard polynomial interpolation remainder formula [4] and Lemma 3.1 to obtain
sup λ∈Λ|g(˜µ, λ) − [eI ˆ u M −1g(·, λ)] (˜µ)| ≤ supλ∈Λ supz∈Z M !1 |D M 1 g(z, λ)| ˜∆M ≤ (2 ˜∆)M˜µ( ˜∆,δ N). (21)
We now assume that c∗δN
2 ≤ ˜∆ and ˜∆≤ 1
2; under these conditions (recall
(18)) we obtain (2 ˜∆)Mµ˜( ˜∆,δ
N) ≤ (2 ˜∆)∆/c˜ ∗
δN, and hence, from (20) and
(21), we can write
|||˜uN(˜µ)− ˆuµ˜
||| ≤ |||uN(0)|||(2 ˜∆)∆/c˜ ∗δN . (22)
It remains to select a best ˜∆ satisfying c∗δN
2 ≤ ˜∆≤ 1 2.
To provide the sharpest possible bound, we choose ˜∆ = ˜∆∗
≡ 1 2e, the
minimizer (over all positive ˜∆) of (2 ˜∆)∆/δ˜ N. Our conditions on ˜∆ are
readily verified: c∗δN
2 ≤ ˜∆
∗ follows directly from the hypothesis of our
lemma, N ≥ Ncrit; and ˜∆∗ ≤ 12 follows from inspection. We now insert
˜
∆ = ˜∆∗ into (22) to obtain
|||˜uN(˜µ)− ˆuµ˜||| ≤ |||uN(0)||| e−N/Ncrit,
∀ ˜µ ∈ eD . It immediately follows that, for any µ∈ D,
inf wN N∈W N N |||uN(µ)− wNN||| = inf wN N∈W N N |||˜uN(τ−1(µ))− wNN|||
≤ |||˜uN(τ−1(µ))− ˆuτ−1(µ)||| ≤ |||uN(0)||| e−N/Ncrit ,
since ˆu·∈ WN
N and, for µ∈ D, τ−1(µ)∈ eD. This concludes the proof.
Then, from (10),(11), Lemma 3.1, and Lemma 3.2, we obtain Theorem 3.3 For N ≥ Ncrit≡ c∗e ln(γµmax+ 1),
|||uN
(µ)− uN
N(µ)||| ≤ (1 + µmaxγ1)1/2|||uN(0)||| e−N/Ncrit, ∀ µ ∈ D;
furthermore,
|||u(µ) − uN (ε)N (µ)||| ≤ ε + (1 + µmaxγ1)|||u(0)||| e−N/N
crit
, ∀ µ ∈ D, forN (ε) such that |||u(µ) − uN (ε)(µ)||| ≤ ε.
4
Conclusions
We make several observations about the results of Theorem 3.3. First, we obtain exponential convergence with respect to N . Second, our con-vergence result applies uniformly for all µ∈ D. Third, our convergence parameter Ncrit depends only very weakly — logarithmically — on γ1,
re-lated to the form of the operator, and on µmax, related to the range of the
parameter (note we may also view the product γ1µmax as the
thus achieve convergence “soon” (N ≥ Ncrit) with a “large” (N1crit)
expo-nential decay rate. Fourth, we obtain convergence with respect to both N andN : uN
N(µ)→ u(µ) as N, N → ∞.
Let us now make several remarks concerning the point distribution. First, the logarithmic point distribution is intimately related to our partic-ular abstract problem, (1), (3), and, relatedly, the parametric dependence of the solution, (6). In brief, for larger values of µ, the derivatives of (1 + µλ)−1 will be smaller, thus permitting a larger interval in which to
recruit the points required for accurate interpolation. Second, it should be clear from the proof of Lemma 3.2 that the requirement on the point distribution is, in fact, rather weak: the location of the M points within e
I∆µ˜˜ is (save for conditioning issues) irrelevant. This permits, for example, log-random distributions — particularly attractive in higher parameter di-mensions in which tensor-product grids are prohibitively costly.
Third, the logarithmic point distribution is not an artifact of our (inter-polant-based) proof: in numerical tests [15] the error in the actual Galerkin approximationis also “minimized” by a logarithmic point distribution; even point distributions that enjoy general optimality properties, such as Cheby-shev, do not perform as well as the logarithmic distribution for our partic-ular problem. (Indeed, for Chebyshev interpolation over the intervalD, it may be shown that Ncritscales as √µmax— much worse than our ln µmax.)
Fourth, we note that numerical tests [15] roughly confirm the depen-dence of|||uN(µ)
− uN
N(µ)||| on N, γ, and µmax. However, in general, our
theoretical bound can be quite pessimistic, as might be expected given that our proof is based on (albeit, tailored) interpolation arguments: Galerkin optimality can always do better, for example, choosing to “illuminate” only an appropriate subsample of SN so as to construct the best
“sub-approximation” (or sub-interpolant) amongst all O(N !) possibilities. This property is no doubt crucial in higher parameter dimensions, in which effec-tive scattered-data higher-order interpolants are very difficult to construct; it is here that the superiority of the reduced-basis approach over simple parameter-space interpolation is most evident. We do not yet have any (uniform in µ) theory for higher parameter dimensions, although numeri-cal results again suggest extremely rapid convergence.
Finally, we note that we address in this paper only one aspect — rapid uniform convergence — of successful reduced-basis approaches. In other papers [6, 7, 8, 12] we focus on (i) off-line/on-line computational decompo-sitions that permit real-time response, and (ii) a posteriori error estimators that ensure both efficiency and certainty.
Acknowledgements. We would like to thank Christophe Prud’homme, Dimitrios Rovas, and Karen Veroy of MIT for sharing their numerical results prior to publication. This work was performed while ATP was an Invited Professor at the University of Paris VI in February, 2001. This work was supported by the
Singapore-MIT Alliance, by DARPA and ONR under Grant F49620-01-1-0458, by DARPA and AFOSR under Grant N00014-01-1-0523 (Subcontract 340-6218-3), and by NASA under Grant NAG-1-1978.
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